(NBC NEWS) -- Senators on a key panel continued slogging through a lengthy list of
proposed amendments to a comprehensive immigration reform bill Tuesday,
even as the legislation's opponents reminded congressional leaders that
they'll continue to fight against "amnesty."

The 18-member Senate
Judiciary Committee unanimously approved an amendment proposed by top
Republican Sen. Chuck Grassley of Iowa to address student visa security
issues highlighted in the aftermath of the Boston Marathon bombing.

That
measure, which requires that student visa information be shared in
real-time with Border Patrol officers at the nation's major ports of
entry, is designed to repair the communication error that led to a
friend of bombing suspect Dzhokhar Tsarnaev entering the United States
despite the expiration of his student visa.

While the panel adopted a handful of proposals by GOP members, other
Republican efforts to change the legislation failed, including one
provision backed by Sen. Marco Rubio of Florida, a Gang of Eight member
and top Republican negotiator on the immigration reform effort.

The
committee rejected an amendment proposed by Sen. Jeff Sessions, a
leading opponent of the bill, that would have required that visa holders
be verified using biometric screening -- like fingerprints or eye scans
-- when exiting the country.

While Democrats said they were
sympathetic to the need for more security at the nation's ports of entry
-- a proposal recommended by the 9/11 Commission Report and mandated by
Congress but never enforced -- senators also warned of an eye-popping
price tag of billions of dollars to implement the system.

Immigration
reform proponents also objected to the amendment's mandate that the
massive security overhaul be in place before undocumented immigrants can
apply for probationary legal status.

That proposal failed 6-12.

Rubio, who is not on the Judiciary
Committee but was an important GOP drafter of the original legislation,
was "disappointed" that the biometric tracking proposal was not adopted
and will fight for it when the bill comes to the Senate floor, a
spokesman said.

"Immigration reform must include the best
exit-system possible because persons who overstay their authorized stay
are a big reason we now have so many illegal immigrants," said Rubio
spokesman Alex Conant in a statement. "We wanted the Judiciary Committee
to strengthen the legislation by adding biometrics to the new exit
system, and we were disappointed by this morning's vote."

Sessions,
who has proposed a total of 49 changes to the bill, failed to win
support from other committee Republicans for a proposal that would have
capped the number of legal immigrants receiving green cards at 1.2
million per year. That measure failed, with all others on the 18-member
panel voting against it.

Sen. Ted Cruz of Texas, who has also been
a vocal critic of the bill as written, argued that the Sessions
amendment would undermine legal immigrants -- a "fundamental pillar of
our country."

Cruz proposed an amendment that would have
dramatically increased caps on visas available for high-skilled foreign
workers, whom he called an "unambiguous good for our country." Bill
drafters -- mindful of the delicate negotiations between business and
labor groups that led to a compromise annual cap of 65,000 -- said the
high maximum could hurt American workers seeking the same jobs. The
amendment fell 4-14.

As the committee's markup negotiations were
under way, a contingent of House Republicans held a press conference a
few blocks away to remind House and Senate leaders that they'll continue
their vocal opposition to a pathway to legal status for undocumented
immigrants.

"There's another viewpoint here" said Iowa Rep. Steve
King, one of the loudest critics of "amnesty" on Capitol Hill. "It is
not the one that's being stampeded in the Senate and may be stampeded in
the House."

King -- who described the immigration bill as "far,
far worse than Obamacare" predicted a rising tide of opposition to the
bill in the Republican-led House -- an assessment echoed by Texas Rep.
Steve Stockman.

Alluding to the eight senators who collaborated
behind closed doors to draft the legislation now being amended in the
Senate, Stockman said that a "Gang of Millions" will make its voice
heard to defeat the sweeping bill.

"The people are stronger than the Gang of Eight," he said.

At
the end of Tuesday's session, the committee had accepted a total of 36
amendments. Over 300 were submitted by the committee's members.